Then substituting FLRW metric:
(assuming that \(\Phi = \Phi (t)\)  and a barotropic perfect fluid \(p = w\rho \)  as a source) one gets the corresponding set of equations for the dynamical variables \(a,\Phi \)  (cf. [1]}):
\(& 3H^2 = \frac{\kappa ^2 N^2\,\rho }{\mathcal {A}(\Phi )} - \frac{3N^2\,K}{a^2} + \frac{1}{2}\frac{\mathcal {B}(\Phi )}{\mathcal {A}(\Phi )}\dot{\Phi }^2 - 3\frac{\mathcal {A}^{\prime }(\Phi )}{\mathcal {A}(\Phi )}H\dot{\Phi } + \frac{N^2\,}{2}\frac{\mathcal {V}(\Phi )}{\mathcal {A}(\Phi )}\,, \\& 2\dot{H} + 3H^2 -2\frac{\dot{N}}{N} H = - \frac{w N^2\,\kappa ^2\rho }{\mathcal {A}(\Phi )} - \frac{N^2\,K}{a^2} -\frac{\mathcal {B}(\Phi ) + 2\mathcal {A}^{\prime \prime }(\Phi )}{2\,\mathcal {A}(\Phi )} \dot{\Phi }^2 \nonumber \\&\qquad \qquad \qquad \qquad - \frac{\mathcal {A}^{\prime }(\Phi )}{\mathcal {A}(\Phi )}\left(2H\dot{\Phi }+ \ddot{\Phi }-\frac{\dot{N}}{N}\dot{\Phi }\right) + \frac{N^2\,\mathcal {V}(\Phi )}{2\,\mathcal {A}(\Phi )}\, ,\\\begin{split}& \left(3(\mathcal {A}^{\prime }(\Phi ))^2 + 2\mathcal {A}(\Phi )\mathcal {B}(\Phi )\right)(\ddot{\Phi } -\frac{\dot{N}}{N}\dot{\Phi }) = -3\left(3(\mathcal {A}^{\prime }(\Phi ))^2 + 2\mathcal {A}(\Phi )\mathcal {B}(\Phi )\right)H\dot{\Phi } \\&\quad \quad \quad \quad -\left((\mathcal {A}(\Phi )\mathcal {B}(\Phi ))^{\prime } + 3\mathcal {A}^{\prime }(\Phi )\mathcal {A}^{\prime \prime }(\Phi )\right)\dot{\Phi }^2 + N^2\left(2\mathcal {V}(\Phi )\mathcal {A}^{\prime }(\Phi ) - \mathcal {V}^{\prime }(\Phi )\mathcal {A}(\Phi )\right) \\& \quad \quad \quad \quad \quad + N^2\,\kappa ^2\rho (1 - 3w)\left[\mathcal {A}^{\prime }(\Phi ) - 2\alpha ^{\prime }(\Phi )\mathcal {A}(\Phi )\right]\,. \end{split}\) 
